====== Rigoberto Flórez (South Carolina at Sumter) ======

====== Projective Representation of Non-Representable Matroids (of Biased Graphs) ======

===== Abstract for the Combinatorics Seminar 2011 May 23 =====

To each quasigroup Q there is a complete graph K<sub>3</sub> with multiple edges corresponding to the elements of Q, with a class **//B//** of selected triangles such that every two non-parallel edges belong to exactly one selected triangle. This is called a "biased expansion" of K<sub>3</sub>, written Q·K<sub>3</sub>. There are two associated rank-3 matroids, the "full frame matroid" G<sup>•</sup>(Q·K<sub>3</sub>) and "extended lift matroid" L<sub>0</sub>(Q·K<sub>3</sub>).

When Q is a subgroup of the multiplicative or additive group of a skew field F, the full frame or extended lift matroid (respectively) is representable in the projective plane over F. Thomas Zaslavsky and I are generalizing this standard theorem to arbitrary quasigroups, the role of F being taken by a planar ternary ring associated with a projective plane. There are complications; for instance, although the skew field associated with a Desarguesian plane is unique, there is not a unique planar ternary ring for a non-Desarguesian plane.


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